Peter Hart - HW 1

PHYS 360

7.3 μs
5.9 ms
euler (generic function with 1 method)
61.9 μs

Problem 2.2.2

47.7 μs

x˙=1x14

xf={1,1}

xfstability
1stable
-1unstable
6.7 μs
40.1 ms
378 ms

Problem 2.2.8

19.4 μs

xf={1,0,2}

dx˙dx|x=1=0dx˙dx|x=0<0dx˙dx|x=2>0

3.4 μs
13.0 μs

x˙=(x(1))2(x0)(x2)

x˙=x(x2)(x+1)2

3.3 μs
53.3 ms

Problem 2.3.2

7.0 μs

x˙=k1axk2x2=x(k1ak2x)

x˙(xf)=0

xf={0,ak1k2}

dx˙dx=k1ak2x

dx˙dx|x=0=k1a>0

dx˙dx|x=ak1k2=k1a2k2(ak1k2)=k1a2k1a=k1a<0

xfstability
0unstable
ak1k2stable
2.8 μs
359 ms

Problem 2.3.3

N˙=aNln(bN)

a)

a is the rate which cells grow/decay

1/b is the maximum number of cells in the tumor, as Nb the ln(bN)0

b)

Let a=1,b=1/10

9.7 μs
326 ms

Problem 2.4.5

x˙=1ex2

xf=0

dx˙dx=2xex2

dx˙dx|x=0=0

xfstability
0half-stable
6.5 μs
44.6 ms

Problem 2.7.1

x˙=dVdx=x(1x)

dVdx=x(x1)=x2x

dV=(x2x)dx

V(x)=x33x22+C

LetC=0

V(x)=x33x22

xfstability
0unstable
1stable
5.8 μs
78.8 ms

Computational Question

x˙=1x2

δ(t)δ(0)=eλt

δx(t)ef(x0)t

λf(x0)

λ=2x0

x0stabilityλ
1stable-2
-1unstable2
7.1 μs
347 ms
m (generic function with 1 method)
22.4 μs
691 ms
11.2 ms

λ1 = -1.9919314957795113

22.4 μs
133 ms
5.0 ms

λ1 = 1.7649634965340262

6.8 μs